Finite Group Databasebeta

← #28378410

Subgroups — #28378410

Order 13952.

304 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16812756C211yesmaximal, characteristic, complemented
2C4×C22⋊#89137C211yesmaximal, characteristic, complemented
3#20295880C212yesmaximal, complemented
4#20295878C212yesmaximal, complemented
5#20295539C211yesmaximal, characteristic
6C22×#11724786—21no—
7C2×C22⋊#89137—21no—
8C2×C4×#89137—21no—
9#16812756—21no—
10#16812756C2212yes—
11(C2×#89137)⋊C4C2212yes—
12C2×#13182030C411yescharacteristic, complemented
13C22×C4×D109C2211yescharacteristic
14#16812684—21no—
15#16812675C2211yescharacteristic
16#16812674C2211yescharacteristic
17#16812673C411yescharacteristic, complemented
18#16812669—21no—
19#16812661—21no—
20#16812659—21no—
21#16812657C411yescharacteristic, complemented
22C22⋊Q8×C109C411yescharacteristic, complemented
23C22×#89137—22no—
24C2×#11724786—22no—
25C2×#11724786—42no—
26C22⋊#89137—24no—
27C4×#89137—42no—
28C23×D109D411yescharacteristic
29#13182030—42no—
30C2×D436—21no—
31C2×D436—21no—
32C2×D436C2×C411yescharacteristic
33C2×C22⋊D109—21no—
34C22×Dic109D411yescharacteristic
35C2×C4×D109—21no—
36C2×C4×D109—42no—
37C2×C4×D109C2×C411yescharacteristic
38C2×C4×D109D411yescharacteristic
39C2×C4×D109D411yescharacteristic
40C2×C4×D109C2311yescharacteristic
41C4⋊Dic109C2×C411yescharacteristic
42#13182008—21no—
43#13182006—21no—
44#13182006—21no—
45#13182006—21no—
46#13182006—21no—
47C4×Dic109—21no—
48C2×Q8×C109C2×C411yescharacteristic
49C22×C4×C109C2×C411yescharacteristic
50C4⋊C4×C109—21no—
51C4⋊C4×C109C2×C411yescharacteristic
52C22⋊C4×C109—21no—
53C23×C109C22⋊C411yescharacteristic
54C22×D109—22no—
55C22×D109—22no—
56C22×D109—21no—
57C22×D109—41no—
58C22×D109C22⋊C411yescharacteristic
59C22×D109C2×D411yescharacteristic
60C2×#89137—24no—
61C2×#89137—42no—
62C2×#89137—42no—
63Q8×C109—41no—
64C2×C4×C109—21no—
65C2×C4×C109—21no—
66C2×C4×C109—41no—
67C2×C4×C109C22⋊C411yescharacteristic
68C2×C4×C109C22×C411yescharacteristic
69C22⋊D109—42no—
70C2×Dic109—22no—
71C2×Dic109—21no—
72C2×Dic109C22⋊C411yescharacteristic
73C2×Dic109C2×D411yescharacteristic
74D436—41no—
75D436—42no—
76D436—42no—
77C4×D109—22no—
78C4×D109—22no—
79C4×D109—41no—
80C4×D109—41no—
81C4×D109—42no—
82#11724786—42no—
83C22×C109—21no—
84C22×C109—21no—
85C22×C109#10412yes—
86C22×C109C2×C22⋊C411yescharacteristic
87C2×D109—21no—
88C2×D109—21no—
89C2×D109—21no—
90C2×D109—21no—
91C2×D109—21no—
92C2×D109—44no—
93C2×D109—41no—
94C2×D109—41no—
95C2×D109—41no—
96C2×D109#13512yes—
97#89137—44no—
98C4×C109—21no—
99C4×C109—21no—
100C4×C109—41no—
101C4×C109—41no—
102Dic109—21no—
103Dic109—41no—
104Dic109#9912yes—
105C2×C109—22no—
106C2×C109C2×#9911yescharacteristic
107C2×C109#41611yescharacteristic
108C2×C109C2×#10411yescharacteristic
109D109—22no—
110D109—41no—
111D109—81no—
112#1501—1091nomaximal, Sylow
113C109#150111yescharacteristic, complemented, Sylow
114#525—1091no—
115C2×C22⋊C8—1091no—
116C4×C22⋊C4—1091no—
117#328—1092no—
118#327—1092no—
119C2×#41—1091no—
120C23×C4—1091no—
121C22×C8—2181no—
122#123—2181no—
123C22⋊Q8—1091no—
124C22⋊Q8#8913711yescharacteristic, complemented
125#121—1091no—
126#121—1091no—
127#120—2181no—
128C4×D4—2181no—
129C4×D4—2181no—
130#117—1091no—
131C2×C22⋊C4—2181no—
132C2×C42—2181no—
133C22⋊C8—1092no—
134C22⋊C8—2181no—
135#95—1092no—
136C24—1091no—
137#41—4362no—
138C2×Q8C2×#8913711yescharacteristic
139C2×D4—1091no—
140C2×D4—2181no—
141C2×D4—2181no—
142C2×D4—2181no—
143C22×C4—1091no—
144C22×C4—1091no—
145C22×C4—1091no—
146C22×C4—1091no—
147C22×C4—1091no—
148C22×C4—4362no—
149C22×C4—2181no—
150C22×C4—2182no—
151C22×C4C2×#8913711yescharacteristic
152C2×C8—4362no—
153C2×C8—2182no—
154C4⋊C4—21no—
155C4⋊C4—1091no—
156C4⋊C4—2181no—
157C4⋊C4C2×#8913711yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—2181no—
160C22⋊C4—2181no—
161C22⋊C4—2181no—
162C22⋊C4—2184no—
163C22⋊C4—2181no—
164C42—4362no—
165C42—2181no—
166C23—1091no—
167C23—1091no—
168C23—4361no—
169C23—2182no—
170C23—2181no—
171C23—2182no—
172C23C22⋊#8913711yescharacteristic
173Q8—41no—
174D4—4361no—
175D4—4362no—
176D4—4362no—
177D4—4362no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—1091no—
182C2×C4—1091no—
183C2×C4—4361no—
184C2×C4—4361no—
185C2×C4—4362no—
186C2×C4—4362no—
187C2×C4—4362no—
188C2×C4—2182no—
189C2×C4—2182no—
190C2×C4—2182no—
191C2×C4—2184no—
192C2×C4—2181no—
193C2×C4C22⋊#8913711yescharacteristic
194C2×C4C22×#8913711yescharacteristic
195C8—4362no—
196C22—21no—
197C22—21no—
198C22—1092no—
199C22—4364no—
200C22—4361no—
201C22—4361no—
202C22—4361no—
203C22—2181no—
204C22—2181no—
205C22—2181no—
206C22—2181no—
207C22—2181no—
208C22#1681276412yes—
209C22C2×C22⋊#8913711yescharacteristic
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—1092no—
215C4—4364no—
216C4—4361no—
217C4—2181no—
218C2—22no—
219C2—8721no—
220C2—4361no—
221C2—2182no—
222C2D109⋊#9911yescharacteristic
223C2#2029601611yescharacteristic
224C2C2×#1681276411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#89137
commutatorC2×C4×C109C22×C4
FrattiniC2×C4C22×#89137
FittingC22⋊Q8×C109C4
socleC22×C109C2×C22⋊C4
prime radical#20295539C2
cosocleC2×C4×D109C23
hypercenterC22⋊Q8#89137
hypocenterC109#1501
LCSG ▷ C2×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C2×#16812756C2semidirect
2. C4×C22⋊#89137C2semidirect
3. #20295880C2semidirect
4. #20295878C2semidirect
12. C2×#13182030C4semidirect
17. #16812673C4semidirect
21. #16812657C4semidirect
22. C22⋊Q8×C109C4semidirect
113. C109#1501semidirect
124. C22⋊Q8#89137semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168127562
C4×C22⋊#891372
#202958802
#202958782
#202955392
#1501109